Reading of mathemaтical expressions
1. х > у «х is greater than у»
2. х < у «х is less than у»
3. х = 0 «х is equal to zero»
4. х ≤ у «х is equal оr less than у»
5. х < у < z «у is greater than х but less than z»
6. xv «х times or х multiplied by y»
7. а + b «а рlus b»
8. 7 + 5 = 12 «seven plus five equals twelve; seven plus five is equal to twelve; seven and five is (аrе) twelve; seven added to five makes twelve»
9. а – b «а minus b»
10. 7 – 5 = 2 «seven minus five equals two; five from seven leaves two; difference between five and seven is two; seven minus five is equal to two»
11. а x b «а multiplied bу b»
12. 5 x 2 = 10 «five multiplied by two is equal to ten; five multiplied by two equals ten; five times two is ten»
13. а : b «а divided bу b»
14. a/b «a over b, оr а divided bу b»
15. 10 : 2 =5 «ten divided by two is equal to five; ten divided by two equals five»
16. а = b «а equals b, оr а is equal to b»
17. b ≠ 0 «b is not equal to 0»
18. т : аb «т divided bу а multiplied bу b»
19. √ах «The square root of ах»
20. ½ «one second»
21. ¼ «one quarter»
22. -7/5 «minus seven fifth»
23. a⁴ «а fourth, а fоurth роwеr оr а exponent 4»
24. аⁿ «а nth, а nth power, or а exponent n»
25. π
e «e to the power π »
26. ⁿ√b «The nthroot of b»
27. ³√8 «The cube root of eight is two»
28. Log ₁₀ 3 «Logarithm of three to the base of ten»
29. 2 : 50 = 4 : x «two is to fifty as four is to x»
30. 4! «factorial 4»
31. (а + b)² = а² + 2аb + b² «The square of the sum of two numbers is equal to the square оf the first number, plus twice the product оf the first and second, plus the square of the seсоnd»
32. (a-b)² =a² - 2ab+ b² «The squаrе ofthe diffеrеnсе oftwo numbers is equal to the square оf the first number minus twice the product of the first and second, plus the square of the second»
33. ∆х «Increment of х»
34. ∆х →0 «delta x tends to zero»
35. ∑ «Summation of ...»
36. dx «Differential of х»
37. dy/dx «Derivative of у with respect to х»
38. d²y/dx² «Seсond derivative of у with respect to х»
39. dⁿy/dxⁿ «nth derivative of у with respect to х»
40. dy/dx «Partial derivative of у with respect to х»
41. dⁿy/dxⁿ «nth partial derivative of у with respect to х»
42. ∫ «Integral of...»
a
43. ∫ «Integral between the limits а and b»
b
44. ⁵√dⁿ «The fifth root of d to the nth power»
45. √a+b/ a – b «The square root of а plus b over а minus b»
46. а³ = logсd «а cubed is equаl to the logarithm of d to the base с»
t
47. ∫ f[S, φ(S)] ds «The integral of f of S and φ of S, with respect to S from τ τ to t»
d²y
48. —— + (l + b (S))y = 0 «The second dеrivаtivе of у with respect to s,
ds²
plus у times thе quantity l plus b of s, is equal to zеrо»
tl
49. Х a-ь =e «Х sub а minus b is equal to e to the power t times l»
50. f (z) = Kаb «f of z is equal to К sub аb»
d²u
51. ——— = 0 «The second partial (derivative) of и with rеspect to t is
dt²
equal to zero»
PART III
ADDITIONAL READING
I. MATRICULATION ALGEBRA
DEFINITIONS
1. algebra is the science which deals with quantities.
These quantities may be represented either by figures or by letters. Arithmetic also deals with quantities, but in Arithmetic the quantities are always represented by figures. Arithmetic therefore may be considered as a branch of Algebra.
2. In Algebra it is allowable to assign any values to the letters used; in Arithmetic the figures must have definite values. We are therefore able to state and prove theorems in Algebra as being true, universally, for all values; whereas in Arithmetic only each particular sum is or is not correct. Instances of this will frequently occur to the student of Algebra, as he advances in the subject.
3. This connection of Arithmetic and Algebra the student
should recognize from the first. He may expect to find
the rules of Arithmetic included in the rules of Algebra.
Whenever he is in a difficulty in an algebraical question,
he will find it useful to take a similar question in Arithmetic with simple figures, and the solution of this simple sum in Arithmetic will often help him to solve correctly his algebraical question.
4. All the signs of operation used in Arithmetic are used in Algebra with the same significations, and all the rules for arithmetical operations are found among the
rules for elementary Algebra. Elementary Algebra, how ever, enables the student to solve readily and quickly many problems which would be either difficult or impossible in Arithmetic.
5. Signs and abbreviations.— The following signs and abbreviations are used in Algebra :—
+ plus, the sign of addition.
− minus, the sign of subtraction.
× into, or multiplied by, the sign of multiplication.
÷ by, or divided by, the sign of division.
~ the sign of difference ; thus, a~6 means the difference between a and b, whichever is the larger.
= is, or are, equal to.
.•. therefore.
6. The sign of multiplication is often expressed by a dot placed between the two quantities which are to be multiplied together.
Thus, 2.3 means 2×3; and a. b means a × b.
This dot should be placed low down, in order to distinguish it from the decimal point in numbers. Thus 3.4 means 3×4; but 3·4 means 3 decimal point 4, that is 3 + ·4.
More often between letters, or between a number and a letter, no sign of multiplication is placed.
Thus 3a means 3 × a; and bcd means b × c × d.
7. The operation of division is often expressed by writing the dividend over the divisor, and separating them by a line.
a
Thus — means a ÷ b. For convenience in printing this line is sometimes
b a
written in a slanting direction between the terms ; thus a/b = —.
b
The words sum, difference, multiplier, multiplicand, product, divisor, dividend, and quotient are used in Algebra with the same meanings as in Arithmetic.
8. Expressions and terms. — Quantities in Algebra are represented by figures and by letters. The letters may have any values attached to them, provided the same letter always has the same value in the same question.
The letters at the beginning of the alphabet are generally used to denote known quantities, and the letters at the end of the alphabet are used to denote quantities whose values are unknown. For example, in the expression ax + by — c, it is generally considered that a, b, and c denote known values, but x and y denote unknown values.
An algebraical expression is a collection of one or more signs, figures, and letters, which are used to denote one quantity.
Terms are parts of an expression which are connected by the signs + or —.
A simple expression consists of only one term.
A compound expression consists of two ormore terms.
Thus a, bc, and 3d are simple expressions; and x + 3yz — 2xy is a compound expression denoting one quantity ; and x, 3yz, and 2xy are terms of the expression.
A binomial expression is a compound expression consisting of only two terms; e.g., a+b is a binomial expression.
A trinomial expression is a compound expression consisting of only threeterms; e.g., a — b + c is a trinomial expression.
A multinomial expression is a compound expression consisting of more than three terms.
Positive terms are terms which are preceded by the sign +.
Negative terms are terms which are preceded by the sign −.
When a term is preceded by no sign, the sign + is to be understood. The first term in an expression is generally positive, and therefore has no sign written before it.
Thus, in a + 2b — 3c, a and 2b are positive terms, and 3c is a negative term.
Like terms are those which consist of the same letter or the same combination of letters. Thus, a, 3a, and 5a are like terms; bc, 2bc, and 6bc are like terms ; but ab and ac are unlike terms.
9. The way in which the signs of multiplication and division are abbreviated or even omitted in Algebra will serve to remind the student of the important rule in
Arithmetic that the operations of multiplication and division are to be performed before operations of addition and subtraction.
For example — 2 × 3 + 4 ÷ 2 — 5 = 6 + 2 — 5 = 3.
A similar sum in Algebra would be
ab + — e.
From the way in which this is written, the student would expect that he must multiply a by b, and divide c by d, before performing the operations of addition and subtraction.
10.Index, Power, Exponent.—When several like terms have to be multiplied together, it is usual to write the term only once, and to indicate the number of terms that have to be multiplied together by a small figure or letter placed at the right-hand top corner of the term.
Thus:—
a² means a.a, or a × a.
a3 means a.a. a, or a×a×a.
a4means a.a.a.a, or a× a× a × a.
a2 is read a square;a3 is read a cube ;a4 is read a to the fourth power,or, more briefly, a to the fourth;a7 is read a to the seventh power, or a to the seventh;andso on.
Similarly, (3a)4 = 3a × 3a × 3a × 3a= 81a4; and ab means that b a's are to be multiplied together.
11. Instead of having several like terms to multiply together, we may have a number of like expressions to multiply together. Thus, (b + c)3 means that b + c is to be multiplied by b + (b + c). This will be explained more fully when the use of brackets has been c, and the product multiplied again by b+c; i.e., (b + c)3 = (b + c) × (b + c)× explained.
12. The small figure or letter placed at the right-hand top corner of a quantity to indicate how many of the quantities are to be multiplied together is called an index,orexponent.This index or exponent, instead of being a number or letter, may also be a compound expression, or, in fact, any quantity; but we, at first, restricts ourselves to positive integral indices. We say, therefore, that an indexor exponentis an integral quantity, usually expressed in small characters, and placed at the right-hand top corner of another quantity, to express how many of this latter quantity are to be multiplied together. A power is a product obtained by multiplying some quantity by itself a certain number of times.
13. Notice carefully that an index or an exponent expresses how many of a given quantity are to be multiplied together. For example, a5 means that five a's are to be multiplied together. In other words, the index expresses how many factors are to be used. The index, if a whole number, is always greater by one than the number of times that the given quantity has to be multiplied by itself. For example, the 5 in a5 expresses the fact that five factors, each equal to a, are to be multiplied together; or, in other words, that a is to be multiplied by itself fourtimes. Thus, a5 = a × a × a ×a × a. This fact is often overlooked by beginners.
14. Factor, Coefficient, Co-Factor.— A term or expression may consist of a number of symbols, either numbers or letters, which are multiplied together. For example, the term 15a2bc consists of the numbers 3 and 5 and the letters a, a, b, c all multiplied together.
A factor(Lat. facere,to make) of an expression is a quantity which, when multiplied by another quantity, makes, or produces, the given expression. In the above example 3, 5, a, b, c, and also 15, ab, ac, &c., are all factors of 15a2bc. For we may consider that
15a2bc = 3 × 5 a × a × b × c;
or that 15a2bc = 15 × ab × ac;
or that 15a2bc = 15 × a2 bc;
or that 15a2 bc = ab × 15ac ; &c.
15. It is evident that the term 15a2 bc may be broken up into factors in several ways. Sometimes the factors of a quantity may be broken up again into simpler factors. Thus the factors 15 and a2 bc may be broken up again into 5 and 3 and into ab and ac; and ab and ac may be broken up again into a and b, and into a and c. When a quantity has been broken up into its simplest factors, these factors are called the simpleor primefactors of the quantity. In whatever way we begin to break up a given integral quantity into factors, if we continue to break each factor into simpler factors as long as this is possible, we shall always arrive at the same set of simple factors from the same integral quantity. There is therefore only one set of simple or prime factors for the same integral quantity. In the above example the simple factors of 15a2bc are 3, 5, a, a, b, c.
16. When a quantity is broker up into only two factors,
either of these factors may be called the Coefficient or
Co-Factorof the other factor. For example, in 15a2bc we may call 15 the coefficient of a2bc, or 15a2 the coefficient of bc, or 3ab the coefficient of 5ac, &c. It is convenient, however, to use the word coefficient in the sense of numerical coefficient, and to speak of 15 as the coefficient of a2 bc in 15a2bc. In this sense the coefficient of a quantity is the numerical factor of the quantity.
17. In Arithmetic the factors of a whole number or
integer are always taken to be wholenumbersor integers.The factors of a fraction may be either integers or fractions. For example, the factors of 6/5 may be either 3 and 2/5, or 2 and 3/5, or 6 and 1/5; or, again, the factors of 3/4 may be taken as ½ and 2/3, or as 3 and 1/4. In the case of fractions, a fraction can be broken up into different sets of simple factors in an infinite number of
ways.
18.The coefficient of a quantity may be either integral or fractional. Thus in 5/6a2b the coefficient is 5/6. When no coefficient is expressed, the coefficient oneis to be understood. Thus ab means onceab, just as in Arithmetic 23 means once23.
19. Roots. — We have seen that a×a = a2. Here we multiply the quantity a by itself and so get a2. Suppose we reverse this process; that is, we have a quantity given us, and we try to find some quantity which, when multiplied by itself, will produce the given quantity. For example, what quantity multiplied by itself will give a2? Evidently, a is the required answer. Again, what number multiplied by itself will produce 16? Here 4 is the answer. In these cases we are said to find a root of a2, and of 16.
A root of a given quantity is a quantity which, when multiplied by itself a certain number of times, will produce the given quantity.
20. The square root of a given quantity is that quantity which, when two of them are multiplied together, produces the given quantity. Thus, the square root of a2 is a; because two a's multiplied together produce a2. Again, the square root of 16 is 4, because two fours multiplied together produce 16.
The square root of a quantity is indicated by the sign √, which was originally the first letter in the word radix, the Latin for root. Thus, √16 = 4 ; √a2= a.
21.The cube root of a given quantity is that quantity which, when three of the latter are multiplied together, produces the given quantity. The cube root of a quantity is indicated by the sign ³√. Thus, ³√64= 4, because 4 × 4 × 4 = 64. Similarly, ³√a3 = a, because a × a × a= a3.
22.In like manner 4√, 5√, 6√ &c., are used to indicate the fourth, fifth, sixth, &c.,
roots of a quantity. Thus, √64 = 2, because 2×2×2×2×2×2 = 64. Similarly,
⁵√a5 = a; 7√a7 = a ; ³√x3= x ; ³√y3 = y.
23.With regard to Square and Cube Boot, the student may notice that in Mensuration, if the area of a square is given, the length of each side of the square is expressed by the square root of the quantity expressing the area. For example, a square whose area is 16 square feet has each side 4 feet long. Similarly, a cube whose content is 27 cubic feet has each edge 3 feet long.
24. Brackets. — In Arithmetic each number, as, for example, 13, is thought of as one number. It is true that 13 is equal to the sum of certain other numbers; e.g.,6 + 4 + 3 = 13; but we do not necessarily consider 13 as made up of these numbers, 6, 4, and 3. So also in Algebra each expression must be considered as expressing one quantity, e.g., a + b — c represents the one quantity which is obtained by adding 5 to a and then subtracting c from the sum of a and b.
So also each of the expressions in Exercises I a and I b represents one quantity. The answer to each of these examples is the numerical value of the example when the letters a, b, c, d, e, and f have the numerical values mentioned.
25. When an expression is made up of terms containing
the signs +, —, ×, and ÷, either expressed or understood, we know from Arithmetic that the operations of multiplication and division are considered as indicating a closer relation than the operations of addition and subtraction. The operations of multiplication and division must be performed first, before the operations of addition and subtraction. For example,
3 + 8 ÷ 2 — 2×3 = 3 + 4—6 = 1.
Exactly the same rule applies in Algebra. For example, consider the expression a + bc — d÷ e + f. Here we must first multiply b by c, and divide d by e. Then we add the product to a, then subtract the quotient, and finally add f to get the result.
26. Frequently, however, it is necessary to break this rule about the order of operations, and we may wish some part of an expression to be considered as forming but one term. This is indicated by placing in brackets that part which is to be considered as one term.
For example, in Arithmetic, (3 + 7) × 2 = 10 × 2 = 20. Here we treat 3 + 7 as one term, and therefore we place it in brackets. If we leave out the brackets,
3 + 7×2 = 3 + 14 = 17. Exactly the same thing is done in Algebra. For example, (a + b) ×c means that the sum of a and b is to be multiplied by c; whereas a+b×c means that first of all b isto be multiplied by c, and then the product is to be added to a.
27.Negative quantities.— In Arithmetic, in questions
involving subtraction, we are always asked to take a smaller quantity from a larger quantity. For example, if we have to find the difference between 5 and 7, we say 7—5 = 2. But suppose we are asked to subtract 7 from 5.
Arithmetically this is impossible. In Algebra such a question is allowable. We say that 5 — 7 = 5—5—2 = —2, and we arrive at a negative answer, namely —2. In Algebra, therefore, we may either subtract 5 from 7, or 7 from 5; and we consider it correct to write a — 5, whether a is larger or smaller than b.
28. Instead of considering abstract numbers like 5 and 7, let us suppose that we have to deal with concrete quantities such as £5 and £7. Suppose a tradesman made a profit of £5 one day, and then lost £7 the next day. How should we express his total profit? We should say £5 —£7 = —£2; his profits on the two days amounted to —£2; or, in other words, he lost +£2. It appears then that the negative result of —£2 profit can be expressed as a positive result of + £2 loss.
29. Again, suppose a ship sails 5 miles towards a harbour, and then is carried back by wind and tide 7 miles away from the harbour. We might say that the ship has advanced (5 — 7) miles, or —2 miles towards the harbour;
or that the ship has retired + 2 miles from the harbour.
30. In both these examples a negative answer can be expressed as a positive answer by altering the form of the answer. This can always be done with concrete quantities, and, in Arithmetic, whenever we arrive at a negative result, we transpose the form of the answer and express the result as a positive answer. In Algebra, however, it is convenient to leave a negative result and even to speak of a negative quantity without expressing any positive quantity. Thus we speak of —a, or of —3b, &c., as well as of +a or of +3b, &c.
31. The signs, therefore, + and — are used to distinguish quantities of opposite kinds. Every term in an algebraical expression and also every factor in every term must be thought of as being preceded by either + or —. If no sign is expressed, the sign + is understood. This use of the signs + and — is so constant and so important that + and — are often spoken of as the signs in an expression, and to change the signs in an expression means to change all + signs to —, and all — signs to +. For example, a + b—c isthe same expression as -a—b+c with the signs changed.
This use of + and — before each term must not be confused with the use of the same signs to mark operations of addition and subtraction.
ADDITION AND SUBTRACTION
32. We know, from Arithmetic, that the operations of addition and subtraction are mutually opposed. If we add to and subtract from the same number some other number, we shall not alter the number with which we started. For example, suppose we start with 7. Add and subtract 3, thus: 7 + 3—3 = 7; adding and subtracting 3 has not altered the 7. This is true in Algebra.
In Arithmetic we can add together two or more abstract numbers and express them more shortly as a single number, thus : 2 + 3 + 5 = 10; but in Algebra we can only add together and express more shortly terms which are alike, thus: 2a + 3a + 5a = 10a. Terms which are unlike cannot be added together; thus a + b + c cannot be expressed in a shorter form.
33. The rules for addition are as follows: —
(1) Only like terms can be added.
(2) Add together all the like terms that are positive and all the like terms that are negative; subtract the smaller of these sums from the larger, and prefix the sign of the larger sum.
Remember that when no numerical coefficient is expressed the coefficient 1 is understood.
34. Subtraction.— If we add together —7 and 20, we get 13. If we subtract + 7 from 20, we get 13. Therefore to subtract + 7 from 20 gives the same result as adding -7 to 20.
Conversely, since +7 added to 20 gives 27, we might infer that —7 subtracted from 20 would give 27, and this would be correct. Hence we can also infer a general rule for subtraction, viz.: — Change the signs of all the terms in the express-ion which has to lie subtracted, and then proceed as in addition.
For example: — Subtract 3a—4b from 8a + 2b.
Set down as in addition, and change the signs in the lower line, thus:
8a +2b
-3a+4b
5a+6b
By adding, we get 5a + 6i as the difference required.
In working subtraction sums, the signs in the lower lines should be changed mentally. The above sum would then appear thus:
8a +2b
3a - 4b
5a+6b.
The actual process of working this sum after setting it down would be as follows: — Begin with the a's, minus 3 and plus 8; the plus is the larger by 5; therefore set down 5a, omitting the plus sign because 5a is the first term. Again, plus 4 and plus 2 give plus 6; therefore set down plus 6b.
35. As in addition, like terms must be arranged under like terms. Take another example. From 5a3 + 5a2 — 7a + 3a4—5 take 5a2 — 6a + 7 — 2a4 + 2a3. Arrange in order thus:
3a4 + 5a3 + 5a2 - 7a – 5
-2a4 + 2a3 +5a2 - 6a+ 7
5a4 + 3a3 — a — 12.
The working of this question was as follows: — Begin with a4 plus 2 and plus 3 give plus 5a4. Then, for a3, minus 2 and plus 5, the plus is the larger by 3; therefore set down + 3a3. Then a2 minus 5 and plus 5 give 0; therefore set nothing down. Then a plus 6 and minus 7, the minus is the larger by 1; therefore set down —a, the 1 being understood before the a. Lastly, minus 7 and minus 5 give —12.
36. Notice that in Algebra we do not consider which expression is the larger in a subtraction sum. The answer may be either a positive or a negative quantity; and so in Algebra we may subtract either a larger quantity from a smaller, or a smaller quantity from a larger. Also the letters used in an algebraical expression may have anyvalue, so that we cannot always tell which, is the larger of two expressions. We usually, therefore, pay no attention in Algebra to the magnitudes of the quantities we use.
37. Since subtraction and addition are inverse operations, we can prove the accuracy of our work in an addition sum by subtracting one or more of the expressions added together from the sum; and we can prove the accuracy of a subtraction sum by adding the expression subtracted to the remainder. To take an example from Arithmetic: 7+5+3=15; to prove that this is correct, we subtract 3 from 15, thus: 15— 3 = 12; then subtract 5 from 12, thus: 12 — 5 = 7; we have now come back to 7, which is the number we started with; so we infer that our addition was correct. In subtraction, 14—5 = 9 and 5 + 9 = 14. This will be evident to the student from his knowledge of Arithmetic.
MULTIPLICATION
38. In Arithmetic we say 2×3 = 6. No notice is taken of signs; but, if this be expressed fully and correctly, we should say + 2× +3 = +6. Therefore, when two terms with plus signs are multiplied together the product is plus.
Suppose +2 × -3 or -2 × +3. Evidently the product will not be the same in either of these cases as in +2 × +3. Therefore we assume that +2 × -3 = -6 and -2 × +3= -6.
Therefore, when one term has a plus sign and the other term has a minus sign the product is minus.
Again, suppose -2 × -3. This is different from the last two cases, and we assume that -2 × -3 = +6. Therefore, when two terms with minus signs are multiplied together the product is plus.
From these results we can infer the rule of signs.
Rule of signs. — Like signs produce plus; unlike produce minus.
39. The application of the rule of signs is very important when we come to deal with indices or powers, and roots of quantities. For example:
(+a)2 =+a × +a =+a2 = a2.
(-a)2 =-a × -a =+a2 = a2.
(+a)3 =+a × +a ×+a =+a3 = a3.
(-a)3 =-a × -a ×-a = -a3.
We see that a plus quantity raised to any power produces a plus result; a minus quantity raised to an even power produces a plus result,e.g., (-a)6 = a6; buta minus quantity raised to an odd power produces a minus result,
e.g., (-a)7 = -a7.
Again, with roots √(a2) = either +a or -a, since + a × +a = a2, and -a × - a = a2 also.
So that the square root of a positive or plus quantity is either plus or minus; that is, every positive quantity which is an exact square has two roots, these roots being of opposite sign — the one plus and the other minus.
Since like signs produce plus, we cannot find the square root of any negative or minus quantity, e.g., √(-a2) is impossible quantity, for - a ×- a = +a2, and
a × a = +a2.
Again, 3√(+a3) = +a3, since +a × +a × +a = +a3; and 3√(-a3) = -a3, since -a × -a × -a = -a3.
We see, therefore, that apparently there is only one real or possible cube root of a given quantity, but this given quantity may be either plus or minus.
Similarly, for higher powers; if we are asked to find the 4th, 6th, 8th, or any even root of a, given quantity, we can only do so when the given quantity is plus, and then we can find two real roots, one of each sign. But, if we are asked to find the 5th, 7th, 9th, or any odd root of a given quantity, we may be able to do so whatever the sign of the quantity is, but we can only find one real root, and the sign of this root will be the same as the sign of the given quantity.
40. These conclusions must be understood to be true only in a limited sense. It is only in a few cases that any root can be obtained exactly; as, for example, the square roots of 4, 9, 16, &c., of a2, a4, a6, &g. ; the cube roots of 8, 27, &o., and of a3, a6, a9, &c. But we can calculate roots of numbers to some required degree of accuracy, or we can express the roots algebraically without actually calculating them, e.g., 5√(a4), 7√(a2), 8√(a3), &g. The student also will learn afterwards to consider that every quantity has just as many roots as the power of the root, e.g., there are 5 fifth roots of any quantity, 6 sixth roots, 7 seventh roots, and so on. One or more of these roots will be real, and the rest only imaginary.
41. We have already seen that when any term is multiplied by itself the product may be expressed in a simple form by the use of an index or power. Thus
a×a=a2; b×b×b=b3; c×c×c×c=c4; and so on. By reversing the process, b4=b×b×b ×b and b×b= b2.
Therefore b4×b2=b×b×b ×b ×b ×b =b6.
Hence we infer that different powers of the same form may be multiplied by writing the quantity with an index equal to the sum of the indices of the multipliers. In the above example, 4 + 2 = 6; therefore b4×b2 =b6.
Similarly, b3×b5 =b8.
Also, since a= a1, a2 × a= a3, a4 × a= a5; and so on.
Also we have seen that when two different terms are multiplied together the product may be expressed by writing the two terms side by side. Thus: a × b = ab; c2 ×d4 =c2d4.
42. In Arithmetic the student knows that, if several numbers have to be multiplied together, the numbers may be taken in any order. For example:
2 ×3 ×4 = 2 ×4 ×3 = 3 ×4 ×2 = 4 ×3 ×2, &c., for each product is equal to 24. So also in Algebra the terms in any product may be taken in any order. So that
abc = acb = bca = bac = cab = cba.
If, therefore, in Algebra we have to multiply together two or more simple factors, we may place the numerical factors all together, and we may gather together any factors which are powers of the same quantity, and apply the rule for the multiplication of indices. For example:
3a2 b3c4 × 2a b2 c3=3×2 × a2× a× b3× b2× c4× c3 =6× a3× b5 × c7=6a3b5c7.
43. In the multiplication of simple expressions like the above, the student will find it advisable to take the numbers first, then the letters in alphabetical order, and, lastly, to apply the rule of signs.
44. Dimension and degree. — If we take a simple expression and write down separately all the letters used as factors of the expression, and if we then count the letters, we obtain the number of the dimensions, or the degree of the term. Thus 3a2 b3 = 3×a×a×b×b×b, is of five dimensions, or of the fifth degree; or 3a2 b3 is of two dimensions in a, and of three dimensions in b. In multiplication, the dimensions of the product must be equal to the sum of the dimensions of the factors. With integral indices, the dimension or degree of any term is equal to the sum of all the indices; thus 3abc2 is of the fourth degree, the indices being 1, 1, 2.
45. The following considerations will enable the student to test the correctness of his work. Notice that —
(1) There are as many lines as there are terms in the multiplier.
(2) There are as -many terms in each line as there are
terms in the multiplicand
(3) With regard to signs, a plus sign in the multiplier
will leave all the signs the same as in the multiplicand. Conversely, a minus sign in the multiplier will change all the signs of the multiplicand
in the corresponding line of the product.
(4) It is advisable to arrange both multiplicand and multiplier in descending powers of some letter, because by so doing we shall find that the products produced in the working will be easier to arrange in columns.
46. A compound expression in which all the terms are of the same dimension is said to be homogeneous.
Since the dimension of every term in a product is equal to the sum of the dimensions of its factors, it follows that, if we multiply together two homogeneous expressions, we shall obtain a homogeneous product.
47. The following rules will therefore enable us to read off the product when two binomial expressions, such as x + 7 and x — 8, are multiplied together.
(1) The first and last terms in the product are obtained by multiplying together the two first terms, and then the two last terms.
(2) The coefficient of the middle term in the product is obtained by adding together, algebraically, the two last terms; e.g.,
(a + 6)(a + 4) = a2 + 10a+24.
Similar rules will hold if, instead of a number, we use any other kind of term for the second term in each multiplier; e.g., (a + 2b)(a + 3b) = a2 +5ab+6b2.
II. Base Two Numerals
During the latter part of the seventeenth century a great German philosopher and mathematician Gottfried Wilhelm von Leibnitz (1646—1716), was doing research on the simplest numeration system. He developed a numeration system using only the symbols 1 and 0. This system is called a base two or binary numeration system.
Leibnitz actually built a mechanical calculating machine which until recently was standing useless in a museum in Germany. Actually he made his calculating machine some 3 centuries before they were made by modern machine makers.
The binary numeration system introduced by Leibnitz is used only in some of the most complicated electronic computers. The numeral 0 corresponds to off and the numeral 1 corresponds to on for the electrical circuit of the computer.
Base two numerals indicate groups of ones, twos, fours, eights, and so on. The place value of each digit in 1101two is shown by the above words (on or off) and also by powers of 2 in base ten notation as shown below.
The numeral 1101two means (1x23)4 + (1x22) + (0x2) + (1x1) = (1x8) + (1x4) + (0x2) + (lxl) = 8 + 4 + + 0 + 1 = 13. Therefore 1101two = 13
| ...23 Eights | 2²⁴ Fours | Twos | Ones |
A base ten numeral can be changed to a base two numeral by dividing by powers of two.
From the above you know that the binary system of numeration is used extensively in high-speed electronic computers. The correspondence between the two digits used in the binary system and the two positions (on and off) of a mechanical switch used in an electric circuit accounts for this extensive use.
The binary system is the simplest5 place-value, power-position system of numeration. In every such numeration system there must be symbols for the numbers zero and one. We are using 0 and 1 because we are well familiar with them.
The binary numeration system has the advantage of having only two digit symbols but it also has a disadvantage of using many more digits to name the same numeral in base two than in base ten. See for example:
476 = 111011100 two
It is interesting to note that any base two numeral looks like a numeral in any other base. The sum of 10110 and 1001 appears the same in any numeration system, but the meaning is quite different. Compare these numerals:
10110 two 10110ten 10110seven
+ 1001 two + 1001ten + 1001 seven
III. CLOSURE PROPERTY
In this lesson we shall be concerned with the closure property.
If we add two natural numbers, the sum will also be a natural number. For example, 5 is a natural number and 3 is a natural number. The sum of these two numbers, 8, is also a natural number. Following are other examples in which two natural numbers are being added and the sum is another natural number. 19+4 = 23 and only 23; 6+6=12 and only 12; 1429+357=1786 and only 1786. In fact, if you add any two natural numbers, the sum is again a natural number. Because this is true, we say that the set of natural numbers is closed under addition.
Notice that in each of the above equations we were able to name the sum. That is, the sum of 5 and 3 exists, or there is a number which is the sum of 19 and 4. In fact, the sum of any two numbers exists. This is called the existence property.
Notice also that if you are to add 5 and 3, you will get 8 and only 8 and not some other number. Since there is one and only one sum for 19+4, we say that the sum is unique. This is called the uniqueness property.
Both uniqueness and existence are implied in the definition of closure.
Now, let us state the closure property of addition.
If a and b are numbers of a given set, then a+ b is also a number of that same set. For example, if a and b are any two natural numbers, then a + b exists, it is unique, and it is again a natural number.
If we use the operation of subtraction instead of the operation of addition, we shall not be able to make the statement we made above. If we are to subtract natural numbers, the resullt is sometimes a natural number, and sometimes not. 11—6 = 5 and 5 is a natural number, while 9—9 = 0 and 0 is not a natural number.
Consider the equation 4—7 = n. We shall not be able to solve it if we must have a natural number as an answer. Therefore, the set of natural numbers is not closed under subtraction.
What about the operation of multiplication? Find the product of several pairs of natural numbers. Given two natural numbers, is there always a natural number which is the product of the two numbers?
Every pair of natural numbers has a unique product which is again a natural number. Thus the set of natural numbers is closed under multiplication.
In general, the closure property may be defined as follows: if x and y are any elements, not necessarily the same, of set A (A capital) and * (asterisk) denotes an operation *, then set A is closed under the operation asterisk if (x*y) is an element of set A.
To summarize, we shall say that there are two operations, addition and multiplication, for which the set of natural numbers is closed. Given any two natural numbers x and y, x + y and x x y are again natural numbers. This implies that the sum and the product of two natural numbers exists. It so happens that with the set of natural numbers (but not with every mathematical system) the results of the operations of addition and multiplication are unique.
It should be pointed out that it is practically impossible to find the sum or the product of every possible pair of natural numbers. Hence, we have to accept the closure property without proof, that is, as an axiom.